The Spectrum of Schrödinger Operators with Poisson Type Random Potential

Abstract

We consider the Schrödinger operator with Poisson type random potential, and derive the spectrum which is deterministic almost surely. Apart from some exceptional cases, the spectrum is equal to [0, ∞) if the single-site potential is nonnegative, and is equal to R if the negative part of it does not vanish with positive probability, which is consistent with the naive observation. To prove that, we use the theory of admissible potential and the Weyl asymptotics.

The spectrum of Schrödinger operators with random \delta magnetic fields

“…Though the proof is similar to those of known results [23], [22], [3], we shall give it here to show the singularity of a ω does not violate the argument.…”

“…So the equation (1.10) tells σ(H ω ) generically fills the whole possible energy range [B, ∞); similar results are found in the case of the Schrödinger operators with the Poisson-Anderson type random scalar potentials [3]. We believe (1.10) holds in general (even if supp α is a finite set of rationals), but it is not yet proved at present.…”

“…The set Γ ω is the Poisson configuration (the support of the Poisson point process) with intensity measure ρdx dy, where ρ is a positive constant (for the definition of the Poisson point process, see e.g. Reiss [31] or Ando-Iwatsuka-KaminagaNakano [3]). The random variables {α γ } γ∈Γω are i.i.d.…”

“…In section 2, we give some basic definitions. In section 3, we introduce the method of admissible potentials by Kirsch and Martinelli [23] (see also Kirsch [22] and [3]). To apply this method, we prove the strong resolvent continuity of our operators with respect to the position parameters γ and the intensity parameters α γ , later in section 7.…”

A Self-adjointness Criterion for the Schrödinger Operator with Infinitely Many Point Interactions and Its Application to Random Operators

“…For this purpose, we use the method of admissible potentials, which is a useful method when we determine the spectrum of the random Schrödinger operators (see e.g. [19,24,5,18]).…”